= Standing modes in an exponentially tapered channel
{title2=$k_n^2=(n\pi/L)^2+\beta^2/4$}
For constant long-wave speed and $S_0\propto e^{\beta x}$, a standing free-surface amplitude obeys $\eta''+\beta\eta'+k^2\eta=0$. The transformation $\eta=e^{-\beta x/2}Y$ gives $Y''+(k^2-\beta^2/4)Y=0$. Impermeable ends at $0,L$ impose $\eta'=0$, yielding the displayed nonzero-frequency eigenvalues and $\eta_n\propto e^{-\beta x/2}[\cos(n\pi x/L)+\beta L\sin(n\pi x/L)/(2n\pi)]$. A separate zero-frequency mode changes the uniform water level.
Back to article page
